AS we can see, higher magnetic fields lead to lower radifrequencies, which means that the energy splitting decreases. This can only be explained if the external magnetic field has an opposite direction with respect to the hyerfine field, in that case, an increase will lower the total magnetic field recived for the atoms, leading to a lower sliting.
The electronic part J doesn’t change between two isotopes. Then, we can write the relation
In our first isotope, we can obtain a1 empirically, and we know I1 as will as mu1.
In our second isotope, we can also calculate a2 empirically, and we know that J will remain the same as in isotope 1, meanhile mu2 and I2 will change. We know I2 value, so if we apply the same magnetic field in both experiments, relating the two expressions we can extract mu2 as a function of a1, a2, I1 and I2, all known values.